(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove T is a linear transformation and find bases for both N(T) and R(T).

2. Relevant equations

3. The attempt at a solution

T:M_{2x3}(F) [itex]\rightarrow[/itex] M_{2x2}(F) defined by:

T(a_{11}a_{12}a_{13})

(a_{21}a_{22}a_{23})

(this is one matrix)

=

(2a_{11}-a_{12}a_{13}+2a_{12})

( 0 0)

(this is one matrix)

So I verified that it is a linear transformation by checking that T(cx+y)=cT(x)+T(y). But I don't understand how to find a basis for the null space and range.

I can see that since N(T)={x:T(x)=0} that N(T) here it all vectors of the form:

(t/2 t -2t)

( b b b)

(this is one matrix)

Since the 2nd row in our domain always goes to 0, the second row is arbitrary, which I represented by b.

How do I find a basis for all multiples of the matrix

t(1/2 1 -2)

( b b b)?

And I'm not even sure on how to start off finding the basis for the range. All help is appreciated. Thanks!

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# Homework Help: Finding Basis of Null Space and Range

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